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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\bf x}^{\prime}={\bf A}\,{\bf x}, \quad {\bf A} \mathrm{~is ~real}.
\end{equation*}
</div>
<p class="continuation">The corresponding eigenvalue problem is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
({\bf A}-r{\bf I}) \vec{\xi}={\bf 0}.
\end{equation*}
</div>
<p class="continuation">Suppose there is a complex eigenvalue <span class="process-math">\(r_1=\lambda+i \mu\)</span> and a corresponding eigenvector <span class="process-math">\(\vec{\xi}^{(1)}={\bf a}+i{\bf b}\)</span> where <span class="process-math">\({\bf a}, {\bf b}\)</span> are real. Then</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
({\bf A}-r_1{\bf I}) \vec{\xi}^{(1)}={\bf 0}.
\end{equation*}
</div>
<p class="continuation">Taking complex conjugate gives</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
({\bf A}-\bar{r}_1{\bf I}) \bar{\vec{\xi}}^{(1)}={\bf 0}.
\end{equation*}
</div>
<p class="continuation">This shows that</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
r_2=\bar{r}_1=\lambda-i\mu
\end{equation*}
</div>
<p class="continuation">is also an eigenvalue and</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\vec{\xi}^{(2)}=\bar{\vec{\xi}}^{(1)}={\bf a}-i{\bf b}
\end{equation*}
</div>
<p class="continuation">is also an eigenvector.</p>
<span class="incontext"><a href="sec6_3.html#p-266" class="internal">in-context</a></span>
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